Select the correct answer.
Which inequality represents all the solutions of $10(3 x+2)>7(2 x-4) ?$
A. $x>-4$
B. $x<-4$
C. $x>-3$
D. $x<-3$
Answer (1)
Expand both sides of the inequality: 14x - 28"> 30 x + 20 > 14 x − 28 .
Combine like terms to isolate x : -48"> 16 x > − 48 .
Divide by the coefficient of x to solve for x : -3"> x > − 3 .
The solution is -3"> x > − 3 , which corresponds to option C: -3}"> x > − 3 .
Explanation
Understanding the Inequality We are given the inequality 7(2x-4)"> 10 ( 3 x + 2 ) > 7 ( 2 x − 4 ) . Our goal is to find the solution set for x .
Expanding the Inequality First, we expand both sides of the inequality: 7(2x-4)"> 10 ( 3 x + 2 ) > 7 ( 2 x − 4 ) 14x - 28"> 30 x + 20 > 14 x − 28
Isolating x Next, we want to isolate x on one side. Subtract 14 x from both sides: 14x - 14x - 28"> 30 x − 14 x + 20 > 14 x − 14 x − 28 -28"> 16 x + 20 > − 28
Further Isolating x Now, subtract 20 from both sides: -28 - 20"> 16 x + 20 − 20 > − 28 − 20 -48"> 16 x > − 48
Solving for x Finally, divide both sides by 16 :
\frac{-48}{16}"> 16 16 x > 16 − 48 -3"> x > − 3
Finding the Correct Option The solution to the inequality is -3"> x > − 3 . Comparing this to the given options, we see that option C matches our solution.
Examples
Understanding inequalities is crucial in various real-life scenarios. For instance, when budgeting, you might use inequalities to determine how much you can spend on different items while staying within your income. If you earn a certain amount per week and have fixed expenses, inequalities can help you calculate the maximum amount you can spend on discretionary items without exceeding your budget. This ensures you manage your finances effectively and avoid overspending.
